Exploration of a Nonlinear World: An Appreciation of Howell Tong's Contribution to Statistics

412 Pages·2009·41.75 MB·English

by  Kung-Sik Chan

Exploration of a
Nonlinear World
An Appreciation of Howell Tong’s Contributions to Statistics
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Exploration of a
Nonlinear World
An Appreciation of Howell Tong’s Contributions to Statistics
edited by Kung-Sik Chan
University of Iowa, USA
World Scientific
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British Library Cataloguing-in-Publication Data
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Front cover design by Anna Tong
Back cover image by Nils Chr. Stenseth
EXPLORATION OF A NONLINEAR WORLD
An Appreciation of Howell Tong’s Contributions to Statistics
Copyright © 2009 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic
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ISBN-13978-981-283-627-4
ISBN-10981-283-627-6
Printed in Singapore.
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Foreword
This volume is very appropriately titled ‘Exploration of a Nonlinear World’, because
this is just the path which Howell Tong has followed these thirty-odd years: the study of
the time series analysis of nonlinear models. I have admired his work and am grateful for
this chance to o(cid:11)er my own tribute, even if to do so also means admitting that I have seen
the subject vanish overmy horizon during those years.
Myownworkinthenineteen-(cid:12)ftieswas(cid:12)rmlyrootedinthelinearworld,whichseemed
challenging enough at the time. My principal contribution (Whittle, 1951) was to obtain
the asymptotic (large n) expression
n (cid:25) f^(!)
+logf(!) d!
2(cid:25) (cid:0)(cid:25)"f(!) #
Z
(moduloconstants)forthelog-likelihoodofasampleofnjointly-normalobservations. Here
f(!)isthe spectraldensityofthe processand f^(!)isitsrawestimate,the periodogram. A
gooddealfollowsfromthisanditsspatialanalogue. Iobtainedthe resultbythe unrigorous
approximation of the Laurent covariance matrix occurring in the likelihood by a circulant
matrix. Thenaturalstatisticalpathisofcoursetoappealtoanautoregressiverepresentation
of the process, and it is just when such a representation exists that the above expression is
valid.
I realised very well the importance and ubiquity of nonlinear phenomena, ecology pro-
viding one fertile source, but gave up hope of any substantial analysis. Nonlinearity was
forced on me observationally, however, when a seiche study (Whittle 1954) revealed the
existence of subharmonics. Two principal ‘periodic’ components appeared (characterised
by peaks of a smoothed periodogram) which could be physically identi(cid:12)ed as distinct se-
iches. However, components also appeared whose period was a sum of integral multiples of
these two basic periods. Some thought suggested a physical mechanism, which was in fact
a threshold model.
Howellbeganhiscareerincontroltheory,butmovedwithinafewyearstoamatureand
individual study of time series and their statistical analysis. By 1978 he had seized on the
thresholdtheme,whichhedevelopedprogressivelytoastudyofnonlinearmodelsgenerally,
developing a coherent theory and practical methods. These are set forth particularlyin his
seminal paper with Lim of 1980, and his book of 1983.
Howell’s instinct to stretch an envelope so as to bring in subtler features manifests
itself repeatedly. In both the linear and the nonlinear case he saw the estimation of order
(dimension)asbeingacrucialproblem. Amuchmoreradicalstepwastoextendthestudyof
nonlinearmodelstothatofchaoticmodels(seehispaperof 1995andhisbooksof1990and
2001). An essential feature of chaotic models is the discontinuous (but highly structured)
dependence of the path upon the initial state value, and it is just this feature which one
would expect to be lost in a stochastic version.
Thereviewpapersinthe presentvolumebringoutHowell’squalitieswith discernment.
Some do so also with charm (Cutler), some with idiosyncrasy(Fomby) and some with sub-
stantial technical muscle. As an example of the latter, Cline gives a very thorough account
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vi Foreword
of the way the properties (stability etc.) of a dynamical system transform under stochasti-
cisation of that system. An, Brockwell and Rosenblatt all see nonlinear time series models
as stochastic versions of dynamical systems, and concur with Howell on the importance
in this context of Richard Tweedie’s 1975 paper. Yao and Lawrance attack the di(cid:14)cult
chaos/randomnessquestion by asking how one would determine from observationswhether
the‘deterministic’partofthegeneratorofarandomprocessisorisnotachaoticgenerator.
Leng et al, Gao and Tj(cid:28)stheim all consider the determination of dimension, the latter in
particularcontrastingthecross-validationmethodwiththemethodsassociatedwithAkaike,
Mallows and Rissanen. Ling addresses the central consistency question.
Howellwasalwaysinterestedinparticularapplications,andweseeGewekeandStenseth
(cid:12)nding continuing interest in the hare/lynx data, and Li and Tsay treating the essential
nonlinearities of the (cid:12)nancial models which are now so important.
The striking feature of Howell Tong’s 150 papers, three books and book contributions
and also his personal exposition and engagement is the continuing freshness, boldness and
spiritof enquirywhichinformthem {indeed, properqualities foranexplorer. Hestands as
the recognised innovator and authority in his subject, while remaining disarmingly direct
and enthusiastic. This collection stands as a tribute to his achievements, although there is
no expectation that these will cease.
References
1. P. Whittle (1951) Hypothesis testing in time series analysis. AlmquistandWicksell,Uppsala.
2. P. Whittle (1954) The statistical analysis of a seiche record. J. Marine. Res. 13, 78-100.
Peter Whittle
Statistical Laboratory
Cambridge University
U.K.
E-mail: [email protected]
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vii
Preface
This festschrift celebrates the sixty-(cid:12)fth birthday of Howell Tong. It is a tribute of
ouradmirationtoHowell’spath-breakingandtirelesscontributionstononlineartime series
analysis. As one of Howell’s students, I have been bene(cid:12)ting from his teaching, friendship
and generosity of ideas. In particular, I learn from Howell the signi(cid:12)cance of nonlinearity
and dynamics in statistics and science, the intertwining themes being the linchpin of much
of Howell’s works.
This volume reprints ten selected papers by Howell and his collaborators. We are
grateful to nineteen colleagues for contributing seventeen reviews of Howell’s works. Their
reviews shed lighton Howell’s contributions and modern, relateddevelopments in statistics
and science. We are indebted to Professor Peter Whittle, FRS, for writing the Foreword
with an illuminating overviewof this festschrift.
Many of us admire Howell’s mastery of the English language. In fact, he is also well
versed in classic Chinese. The Tang-style Chinese poem (calligraphy by Mr. Yee-Kwong
Kwan) at the end of the book is written by Howell. We are thankful to Mr. Kwan for the
beautiful calligraphy and his elegant English translation of Howell’s poem.
We thank Howell for providing many valuable photographs which enliven this volume,
and provide a glimpse of the friendship and joy in the community of scholars. We are
thankful to Carol Chan, whose creative editorial design of the photographs enhances the
presentationof thesememorableimages. Wethank AnnaTongforherartisticbookdesign,
a design that was inspired by Howell’s poem. We are grateful to Nils Chr. Stenseth for the
lynx photo as the back-coverof the book.
Kung-Sik Chan
Department of Statistics and Actuarial Science
University of Iowa
Iowa City, Iowa 52242
U.S.A.
E-mail [email protected]
March 1, 2009
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Acknowledgements
Grateful acknowledgementsare made to the following:
TheRoyalStatisticalSocietyforpermissiontoreprintthepapers\ThresholdAutoregression,
Limit Cycles and Cyclical Data (with Discussion)", \On Consistent Nonparametric Order
Determination and Chaos",\On the Analysis of Bivariate Non-StationaryProcesses",\On
Likelihood Ratio Tests for Threshold Autoregression" and \An Adaptive Estimation of
Dimension Reduction Space (with Discussion)".
The Applied Probability Trust for permission to reprint the paper \On the Use of the De-
terministic Lyapunov Function for the Ergodicity of Stochastic Di(cid:11)erence Equations".
The Scandinavian Journal of Statistics for permission to reprint the paper \A Personal
Overview of Non-linearTime Series Analysis from a Chaos Perspective(with Discussion)".
StatisticaSinicaforpermissiontoreprintthepapers\BirthofThresholdTimeSeriesModel"
and \Strong Consistency of the Least Squares Estimator for a Non-ergodic Threshold Au-
toregressiveModel".
The AAAS for permission to reprint the paper \Common Dynamic Structure of Canada
Lynx Populations within Three Climatic Regions" and to reproduce Figure 4 on p. 370.
The National Academy of Sciences of the United States of America for permissions to re-
produce Figures 1 to 3 on p. 366, 368 and 369.